Stochastic Quantization of Relativistic Theories

Abstract

It was shown recently that stochastic quantization can be made into a well defined quantization scheme on (pseudo-)Riemannian manifolds using second order differential geometry, which is an extension of the commonly used first order differential geometry. In this letter, we show that restrictions to relativistic theories can be obtained from this theory by imposing a stochastic energy-momentum relation. In the process, we derive non-perturbative quantum corrections to the line element as measured by scalar particles. Furthermore, we extend the framework of stochastic quantization to massless scalar particles.

Analytic Continuation of Stochastic Mechanics

“…We note that the definition for the second order part deviates by a factor 2 compared to the definition used in our previous works[20,28], see also previous footnote.…”

“…Since the early work by Fényes and Nelson, the theory of stochastic mechanics and its associated stochastic quantization scheme have been extended to include spin [13][14][15], to describe processes on (pseudo-)Riemannian manifolds [13,14,[16][17][18][19][20], and to relativistic theories [12,[21][22][23][24][25][26][27][28]. Furthermore, field theoretic extensions have been made [12,21,25,26,[29][30][31][32][33][34], but this theory is still in its infancy.…”